What is the name of this item and how to get it back?
It looks like I have disabled all manipulators like this:



in which case pressing Ctrl+Space does not have an effect.

Turning on at least one of them like this:



brings the manipulator back.

Hovering over the item does not show a keyboard shortcut, so it must have been a mis-Shift-click instead of an unintended keyboard shortcut.